Breaking the Orthogonality Barrier in Quantum LDPC Codes

Classical low-density parity-check (LDPC) codes are a widely deployed and well-established technology, forming the backbone of modern communication and storage systems. It is well known that, in this classical setting, increasing the girth of the Tanner graph while maintaining regular degree distributions leads simultaneously to good belief-propagation (BP) decoding performance and large minimum distance. In the quantum setting, however, this principle does not directly apply because quantum LDPC codes must satisfy additional orthogonality constraints between their parity-check matrices. When one enforces both orthogonality and regularity in a straightforward manner, the girth is typically reduced and the minimum distance becomes structurally upper bounded. In this work, we overcome this limitation by using permutation matrices with controlled commutativity and by restricting the orthogonality constraints to only the active part of the construction, while preserving regular check-matrix structures. This design circumvents conventional structural distance limitations induced by parent-matrix orthogonality, and enables the construction of quantum LDPC codes with large girth while avoiding latent low-weight logical operators. As a concrete demonstration, we construct a girth-8, (3,12)-regular $[[9216,4612, \leq 48]]$ quantum LDPC code and show that, under BP decoding combined with a low-complexity post-processing algorithm, it achieves a frame error rate as low as $10^{-8}$ on the depolarizing channel with error probability $4 %$.

💡 Research Summary

The paper tackles a fundamental obstacle in the design of quantum low‑density parity‑check (LDPC) codes: the orthogonality constraint H_X H_Z^T = 0 required for CSS constructions. In classical LDPC codes, increasing the Tanner‑graph girth while keeping regular degree distributions yields both excellent belief‑propagation (BP) decoding performance and large minimum distance. However, in the quantum setting the same principle fails because the two parity‑check matrices must be mutually orthogonal, which typically forces a reduction in girth and imposes structural upper bounds on the code distance.

The authors propose a novel construction paradigm that relaxes orthogonality to the active part of the matrices while deliberately allowing the latent (deleted) rows to be non‑orthogonal. They start from two square block‑circulant parent matrices (\hat H_X) and (\hat H_Z). The top J block rows become the active matrices (H_X) and (H_Z); the remaining rows form the latent matrices (\tilde H_X) and (\tilde H_Z). Conventional designs enforce full orthogonality of the parent matrices, which inevitably makes every latent row belong to the code space of the opposite type and, if not also orthogonal to the dual space, turns low‑weight latent rows into logical operators. Consequently the minimum row weight L becomes an upper bound on the quantum distance.

To avoid this, the paper introduces three key ideas:

1. Partial Orthogonality – Only the active matrices must satisfy (H_X H_Z^T = 0). The cross‑terms (H_X \tilde H_Z^T) and (H_Z \tilde H_X^T) are required to be non‑zero, ensuring that low‑weight latent rows are not automatically logical operators.

2. Δ‑Based Commutativity – The parent matrices are built from permutation blocks (F_i) and (G_j). Because the block‑circulant structure depends only on index differences, the authors define a set (\Delta = {(k-i) \bmod L/2 \mid 0\le i,k < J}). They prove (Theorem 3.1) that if for every (r\in\Delta) and every (u) the blocks (F_u) and (G_{r-u}) commute, then the active orthogonality holds. This condition is far weaker than requiring all (F)’s and (G)’s to commute.

3. Ensuring Latent Non‑Orthogonality – Theorem 3.2 shows that to keep the latent rows non‑orthogonal it is necessary that the block size (L) satisfies (L \ge 4J). Moreover, Theorem 3.3 states that there must exist at least one difference (r) outside (\Delta) for which the commutator (\Psi_r) is non‑zero; otherwise the latent rows would become orthogonal as well.

The construction uses Affine Permutation Matrices (APMs), which are generalizations of circulant permutation matrices. An APM corresponds to an affine map (f(x)=ax+b) over (\mathbb{Z}_P); its matrix is denoted (P(f)). Commutativity of two APMs reduces to a quadratic congruence \